In the figure shown,a mass $m$ is attached to a light string which is wrapped around a solid cylinder of mass $M$ and radius $R$. The system starts from rest at $t = 0$. If friction is negligible,what will be the angular velocity at time $t$?

$A$ disc of mass $m$ and radius $r$ is free to rotate about its centre as shown in the figure. $A$ string is wrapped over its rim and a block of mass $m$ is attached to the free end of the string. The system is released from rest. The speed of the block as it descends through a height $h$ is:

$A$ tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega$. The force exerted by the liquid at the other end is

$A$ flywheel having mass $3 \ kg$ and radius $5 \ m$ is free to rotate about a horizontal axis. $A$ string having negligible mass is wound around the wheel and the loose end of the string is connected to a $3 \ kg$ mass. The mass is kept at rest initially and released. The kinetic energy of the flywheel when the mass descends by $3 \ m$ is . . . . . . $J$. $(g = 10 \ m/s^{2})$

$A$ binary star system consists of two stars,one of which has double the mass of the other. The stars rotate about their common centre of mass:

$A$ disc rotating about its axis from rest,acquires the angular speed $100 \, rev/s$ in $4 \, s$. The angle rotated by it during these four seconds (in radian) is ...... $\pi$.